Fine-tuning in the 2HDM

The Two-Higgs Doublet Model (2HDM) is one of the most popular and natural extensions of the Higgs sector; but it has two potential fine-tuning problems, related to the electroweak (EW) breaking and the requirement of alignment with the SM Higgs boson. We have quantified the fine-tunings obtaining analytical expressions, both in terms of the initial 2HDM parameters and the physical ones (masses, mixing angles, etc.). We also provide simple approximate expressions. We have taken into account that the fine-tunings are not independent and removed the “double counting” by projecting the variations of the alignment onto the constant- $$v^2$$ v 2 hypersurface. The EW and the alignment fine-tunings become severe in different, even opposite, regions of the parameter space, namely in the regimes of large and small extra-Higgs masses, respectively; emerging an intermediate region, $$500~{\textrm{GeV}} {\mathop {{}_\sim }\limits ^{<}}\{m_H, m_A, m_{H^\pm }\} {\mathop {{}_\sim }\limits ^{<}}700~{\textrm{GeV}}$$ 500 GeV ∼ < { m H , m A , m H ± } ∼ < 700 GeV , where both are acceptably small. We also discuss a remarkable trend that is not obvious at first glance. Namely, for large $$\tan \beta $$ tan β both the EW and the alignment fine-tunings become mitigated. In consequence, the 2HDM becomes quite natural for $$\tan \beta \ge {\mathcal {O}} (10)$$ tan β ≥ O ( 10 ) , even if $$m_H, m_A, m_{H^\pm }$$ m H , m A , m H ± are as large as 1500 GeV. We explain why this is not the case for the 2HDM stemming from supersymmetry. We have illustrated all these trends by numerically analyzing several representative scenarios.